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Can someone just give me a simple example and simple explanation in words about what the difference is between pairwise and mutual independence?

I have this definition:

Three events $A,B,C$ are mutually independence iff the following conditions hold:

1) the events are pairwise independent

2) $P(A\cap B\cap C) = P(A)P(B)P(C)$

Gerry Myerson
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user130306
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4 Answers4

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I think the difference is best explained by the following example where $A$, $B$ and $C$ are pairwise independent but not mutually independent.

In the example we take $A=[0,1/2]$, $B=[1/4,3/4]$, $C=[0,1/4]\cup[1/2,3/4]$ using the Lebesgue measure, i.e., the probability is equal to its length: $P(A) = 1/2$, $P(B)=1/2$, $P(C) = 1/2$. Then, it is not hard to check that $$ P(A\cap B) = P([1/4,1/2]) = 1/4 = 1/2\cdot1/2 = P(A)\cdot P(B)$$ $$ P(B\cap C) = P([1/2,3/4]) = 1/4 = 1/2\cdot1/2 = P(B)\cdot P(C)$$ $$ P(A\cap C) = P([0,1/4]) = 1/4 = 1/2\cdot1/2 = P(A)\cdot P(C)$$ (thus the events are pairwise independent) and $$ P(A\cap B\cap C) = P(\varnothing) = 0 \neq 1/2\cdot1/2\cdot 1/2 = P(A)\cdot P(B)\cdot P(C)$$ (thus the events are not mutually independent).

Stan Tendijck
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Let $S=\{\,100,010,001,111\,\}$, let $A$ be the event "1st bit is 1", let $B$ be the event "second bit is 1", let $C$ be the event "third bit is 1". Each event has probability one-half, and each pair of events has probability one-fourth, so we have pairwise independence. But the probability of all three events is one-fourth, not one-eighth, so we don't have mutual independence.

Gerry Myerson
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  • See also https://math.stackexchange.com/questions/2184244/example-of-pairwise-but-not-mutually-independent and https://math.stackexchange.com/questions/2974849/on-mutual-independence-and-pairwise-independence and https://math.stackexchange.com/questions/1143256/what-is-an-example-of-pairwise-independent-random-variables-which-are-not-indepe and probably many more. – Gerry Myerson Mar 05 '19 at 00:40
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Pairwise independent $$P(A\cap B) = P(A)P(B)$$ $$P(B\cap C) = P(B)P(C)$$ $$P(C\cap A) = P(C)P(A)$$

If all three of these independence tests are passed, A B and C are pairwise independent. If all three are passed and $$P(A\cap B\cap C) = P(A)P(B)P(C)$$ then A B and C are mutually independent.

Mutual independence implies pairwise independence but Pairwise independence does not imply mutual independence as Stan and Gerry’s answers illustrate.

Martin Hansen
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The intuitive meaning is this:

  • $A,B,C$ are pairwise independent if knowledge about one event gives you no knowledge about any other one event. However, it can be the case that knowledge of a pair of events tells you about the other event, or knowledge of one event tells you something about the combination of the other two events.

  • $A,B,C$ are mutually independent if knowledge of any subset of the events given you no knowledge of any of the events outside the subset.

For example, flip three coins, and let

  • $A=\{$first flip is heads$\}$,
  • $B=\{$second flip is heads$\}$,
  • $C=\{$the two flips are equal$\}$.

These are pairwise independent; knowing $A$ tells you nothing about $B$, of course, but less obviously, knowing $A$ tells you nothing about $C$. However, knowing $A$ and $B$ tells you what $C$ is. Also, if you knew only $A$, you would know something about the combination of $B$ and $C$; if you knew $A$ occurred, you know that $B$ and $C$ either both occur or both do not occur, though you do not know $B$ and $C$ individually.

Mike Earnest
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